Adaptive Linearizing Control of a Nonlinear Chemical Process Based

Ind. Eng. Chem. Res. 2002, 41, 5247-5261

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Adaptive Linearizing Control of a Nonlinear Chemical Process Based on Reduced Design Model Kap-Kyun Noh* and En Sup Yoon School of Chemical Engineering, Seoul National University, Seoul 151-742, Korea

In this paper, a control synthesis method requiring the least knowledge on a given process, such as the available outputs and a parametrized reduced design model, is presented. The reduced design model induced by the state transformation including independent output functions is described in the output variables as the transformed state and its minimal dimension is equal to the relative order of the system. It also has parametric uncertainties mainly resulting from the couplings with the disturbance model, which comprises the rest of the transformed system but is excluded from the design work. The method referred to as the ALIC (adaptive linearizing integral control) is a combination of the time-variant input-output linearization with generic model control-like parameter estimation for asymptotic linearization of the parametric design model. Thus, the method does not need the full state availability and the necessity of exact cancellations of nonlinear terms known as critical weak points of the state feedback linearization. Its applicability and effectiveness are demonstrated by simulation for typical control problems in the chemical processes. And, the simplicity and the clarity of the design procedure are elucidated. 1. Introduction For last two decades, significant developments have been made in the nonlinear systems theory evolved from differential geometry.1 In research work, state feedback linearization renders the nonlinear system to be an equivalent linear one in either a state space sense2,3 or an input-output sense4,5 by means of coordinate transformation and state feedback. It has turned out to be a powerful tool for the control synthesis of nonlinear systems and it has been applied to diverse processes.6-8 Compared with the state space linearization, the inputoutput linearization requires rather mild conditions for its applications and it is appealing in the sense that the usual control objectives such as regulation and tracking are formulated through the output variables of the system. The input-output linearization approach also has been widely accepted in the nonlinear control of chemical processes. The globally linearizing control (GLC) 9 in which nonlinear state feedback makes the inputoutput relationship linear and an external linear controller around the input-output linearized system is used was developed. It was extended to a time-varying nonlinear system10 and to systems with measurable disturbances,11 and it was experimentally implemented to polymerization reactors.12,13 The nonlinear internal model control (NIMC)14 is an extension of a linear internal model control (IMC) to a nonlinear system through an interpretation of the input-output linearization within the IMC structure. Meanwhile, when nonlinear systems such as chemical processes are considered, modeling uncertainties and unknown disturbances are unavoidable. Since the state feedback linearization requires an exact model, the linearizing control law based on the nominal model no longer linearizes the actual process in any sense. To cope * To whom correspondence should be addressed. Tel: 822-873-2605.Fax: 82-2-884-0530.E-mail: [email protected].

with these uncertainties and disturbances, the GLC needs an external controller with an integral term such as a PID controller and the NIMC has an integral property imbedded into the control structure. Various methods to directly take into account uncertainties and disturbances have been proposed, and they can be classified into two categories: adaptive control method and robust control method. When there are parametric uncertainties in the nonlinear terms, parameter estimation as a technique to make asymptotic cancellations of nonlinear terms can be linked with the linearizing control law, which leads to an adaptive control method.15-18 In contrast, the robust control method is to incorporate various uncertainties and disturbances into the state model in terms of bounded perturbations and then nonlinear methods accounting for the perturbations are addressed within an input-output linearization approach.19-22 Concerning the state needed at the state feedback and parameter estimation, most methods simply assume the full state availability, which is infeasible in practice, while the GLC,9 the NIMC,14 and the adaptive IMC16 use the process model as a state observer, which is exposed to uncertainties, disturbances, and incorrect initials and so may provide a largely deviated state from an actual one. A nonlinear state observer as an alternative is another challenging problem, and furthermore, the separation principle no longer holds for the nonlinear systems. The methods previously stated are rather sophisticated for practical use, and any one does not provide a practical solution due to the limitations pertaining to each method. In this paper, a simple and practical control synthesis method requiring the least process knowledge such as the available outputs and a parametrized reduced model will be proposed within an inputoutput linearization approach. The design works proceed on a time-varying parametric uncertain nonlinear model in which uncertain parameters allow for diverse

10.1021/ie011042e CCC: $22.00 © 2002 American Chemical Society Published on Web 09/20/2002

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uncertainties and disturbances and whose description in the output variables as the state eliminates the necessity of the state and of which the dimension is the relative order of the system. And, the generic model control23 (GMC)-like parameter estimation is used as a method to make robust an input-output linearization of the parametric reduced model. In section 2, the reduced design model will be formulated and the applicability of the proposed method will be examined. In section 3, an input-output linearizing integral control and a GMC-like parameter estimation method as methodological tools will be presented. And, in section 4, the effectiveness of the adaptive linearizing integral control (ALIC) as a combination of the input-output linearizing integral control GMC-like parameter estimation will be demonstrated via simulations for typical chemical process control problems and finally, some conclusions will be stated in section 5. 2. Formulation of Design Model and Disturbance Model Consider a control-affine nonlinear system

x˘ ) f(x) + g(x)u y ) [yc,yTs ]T ) [hc(x),hTs (x)]T ) H(x)

(1)

where x ∈ Γ ⊂ Rn, u ∈ Ω ⊂ R, and y ∈ Rq are the state vector, the control input, and the augmented output vector as a collection of the secondary outputs vector, ys ∈ Rq-1, and the controlled output, yc ∈ R, respectively. The vector fields, f(x) and g(x), and the scalar fields, hc(x) and hsi(x), i ) 1, .., q - 1 in the outputs map, H(x) ∈ Rq are smooth on Γ. Γ is a connected, physically feasible, and bounded open set, and Ω is an admissible bounded control input set. Note that the system is a single-input single-output (SISO) system since the controlled output for regulation or tracking is single. And, it is assumed that (a) the output functions in the outputs map, H(x), are independent each other and available and (b) the system is input-output linearizable. Definition. The relative order of the controlled output, yc with respect to the input, u is the smallest integer, r such that

LgLfr-1hc(x) * 0

∀x ∈ Γ

(2)

The relative order means that the control input explicitly appears at the rth derivative of the controlled output and a sort of nonlinear coefficient before the control input is bounded away from zero, which guarantees the input-output linearizability of SISO systems.1,9 Due to the independence of the outputs map, H(x), the following state transformation can be taken

z)

[]

[ ]

H(x) y ) Φ(x) ) η P(x)

(3)

where η ∈ Rn-q is a new state vector and P(x) ∈ Rn-q consists of (n-q) scalar functions keeping the local nonsingularity of the state transformation. The new state, η, often can be taken as (n-q) simple states, and at that case, each row of P(x) as a constant matrix has only one nonzero term equal to 1. By such a state transformation, the output variables become the states

[ ]| [ ]| [ ][ ]

of the transformed system given below

LfH(x) f (x) y˘ z˘ ) ) q+1 η˘ l fn(x)

[]

x)Φ-1(y,η)

LgH(x) g (x) + q+1 l gn(x)

LfH(y,η)

u)

x)Φ-1(y,η)

LgH(y,η)

fq+1(y,η) g (y,η) + q+1 u (4) l l fn(y,η) gn(y,η)

Here, LfH(x) and LgH(x) are the Lie derivatives of each scalar field, Hi(x) along the vector fields, f and g. fi and gi with subscript i are components of each vector field, f and g, appearing at the equation of simple state, xi involved in the state transformation. The bar over functions denotes the corresponding functions expressed in the transformed state through the inversion of the state transformation. As seen in (4), according to the availability of the state, the transformed system can be decomposed into two subsystems; the one referred to as the design model is described in the output variables and the other referred to as the disturbance model is described in unavailable original states. The design work will proceed based on the design model while the disturbance model will be excluded from the design work, but its effects on the design model will be taken into account as parameters in the design model. Assumption 1. The design model can be linearly parametrized in terms of parameters representing couplings with the disturbance model: p

y˘ )

p

fyi (y)θi(t) + {∑gyi (y)θi(t)}u ∑ i)0 i)0 yc ) CTy

(5)

Here, fyi (y) ∈ Rq and gyi (y) ∈ Rq are known vector fields depending on only output variables. Introduced parameters, θi(‚) are time varying and unknown since they depend on the unavailable states of the disturbance model and they may incorporate unknown process parameters and external disturbances. The dependence of the parameters is ignored, and only their timevarying feature is indicated by the argument of time. Specifically, θ0 ) 1 and C ) [10...0]T ∈ Rq. The controller will be synthesized on the design model of (5) as a time-varying parametric uncertain nonlinear system. For the design model to be useful for control synthesis, it is desired that the structural characteristic, such as the relative order, of the original system should be kept and well-defined within the design model. It means that the controlled output is also controllable through the reduced design model and the design model is input-output linearizable. To achieve this, some requirements will be imposed on the available outputs set to be included in the state transformation. The outputs such that the assumption below may be satisfied can be physically measured or numerically estimated. Assumption 2. For the design model of (5), (a) the relative order of the augmented output vector, y, with

Ind. Eng. Chem. Res., Vol. 41, No. 21, 2002 5249

respect to the input, u, is 1; i.e., the smallest integer, k, is 1 such that

∂y(k) i *0 ∂u

∀[y,η]T ∈ Φ(Γ), 3i ∈ [1,..,q]

(6)

(b) The relative order of the controlled output, yc, with respect to the input, u, is well defined, and it is equal to that of the original system. Assumption 2(a) means that the input appears explicitly among the first derivatives of available outputs and that at least one of the output functions in H(x) depends on the states directly affected by the input, and nonlinear terms before the input are well defined. Assumption 2(b) is for controllability of the controlled output through the design model, and it requires that at least r appropriate independent outputs be available because the relative order is the minimal dimension of the model for input-output linearizability and the output variables are the states of the design model. For the well-posedness of the relative order of the design model, one should be cautious of introducing the parameters of (5) because a chain of r integrators of the controlled output should be concatenated by the output variables and the dependence of parameters on the outputs is ignored and the parameters are treated just as time-varying ones. Since the relative order is a useful notion to reveal internal structural characteristics of the nonlinear systems and it imposes a restriction on the number of necessary outputs, the applicability of the assumptions to a system can be examined according to the relative order of the system. Case 1. r ) 1: When the system has the relative order of 1, the control input appears at the first derivative of the controlled output. Regardless of the availability of secondary outputs, assumption 2 is trivially met. The number and the details of uncertain parameters in the design model depend on how many secondary outputs are available, and the estimability of introduced parameters complies with the sufficient condition for parameter estimation in section 3.2. The availability of secondary outputs gives much flexibility for the satisfaction of assumption 1 since the number of estimable parameters is allowed up to the number of available outputs. This fact is also valid for case 2 below. Case 2. 2 e r < n: Unlike case 1, at least (r - 1) secondary outputs satisfying assumption 2 are required. Necessary outputs may be easily measurable or some of them may be obtained through new sensor installations or numerical soft-sensors. Once the relative order is defined within the design model with an introduction of appropriate parameters, this case is reduced to case 1. Detailed examples will be illustrated in section 4. Case 3. r ) n: When the relative order is n, n independent outputs should be available, and in contrast to cases 1 and 2, the disturbance model is not formed. The design model is another system equivalent to the original system but described in the output coordinates. Control synthesis based on the design model is the same as the usual state feedback design. In this case, the parameters introduced due to unknown states do not exist.

3. Input-Output Feedback Linearization and Parameter Estimation Based on Reduced Design Model 3.1. Input-Output Feedback Linearization. The input-output state feedback linearization for both a time-invariant and a time-variant nonlinear system has been well established.1,9,10 Brief review and its adaptation to the design model will be done at this subsection. The design model of (5) can be rewritten as a general time-variant nonlinear system:

y˘ ) f y(y,t) + gy(y,t)u yc ) CTy ) hy(y)

(7)

p p where f y(y,t) ) {∑i)0 f yi (y)θi(t)} ∈ Rq and gy(y,t) ) {∑i)0 y gi (y)θi(t)} ∈ Rq are vector fields on H(Γ) × R, C ) [1,0,..,0]T, and hy ∈ R is a scalar field on H(Γ). Since the design model is a time-variant system, the relative order of (2) has to be defined by the modified Lie derivatives having explicit time dependence, but its meanings and roles in the linearization approach are changeless. The existence of a finite relative order ensures the locally invertible state transformation and the state feedback linearizing the input-output response.10 The normal form1 in the error coordinate and with an integrator of the output error is derived for convenience at the output tracking problem and for free offset at the controlled output:

σ˘ ) e1 e˘ 1 ) e2 : e˘ r-1 ) er r y r-1 y e˘ r ) {-y(r) sp + (Lf yh )(y,t) + (LgyLf y h )(y,t)u}

(8a)

j r+1(e1 + ysp,..,er + y(r-1) η˘ r+1 ) η sp ,η) : η˘ n ) η j n(e1 + ysp,..,er + y(r-1) sp ,η)

(8b)

(i-1) y where ejr ) [e1,..,er]T ∈ Rr with ei ) (Lfi-1 is y h )(y,t) - ysp the error coordinate vector and σ as a state is an integrator of the output error, e1. Here, the modified Lie derivatives defined below are used:

(Lf0 yhy)(y,t) ) hy(y) y y (Lfk yhy)(y,t) ) 〈dLk-1 f y h , f 〉 (y,t) +

∂ k-1 y (L h )(y,t), ∂t f y k ) 1,2,.. (9)

where 〈,〉 means an inner product between two vectors. The η ∈ Rn-r is a new state vector and η j (‚) ∈ Rn-r is nonlinear maps, which originate from the excluded disturbance model, but are included for clarity and completeness. The y(i-1) is the (i - 1)th derivative of sp the reference output, ysp, which is assumed to be continuously differentiable as many times as necessary. The state feedback law linearizing between the new input, ν, and the output error, e1, is evident from the

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normal form:

u)

r y(r) sp - (Lf yh)(y,t) + v y (LgyLfr-1 y h )(y,t)

(10)

with the new input, ν taken as r

v)

∑ Rkek + R0∫t k)1

t

0

e1 dt′

(11)

The resulting linear error system has the characteristic polynomial of (r + 1)th order with coefficients Ri, i ) 0,1,..,r and the coefficients, Ri, are chosen to meet the desired feedback properties of the system. The feedback linearization severely depends on exact cancellations of nonlinear functions. To achieve the asymptotic linearization of the design model as a parametric uncertain system, the linearizing control law of (10) and (11) can be linked with parameter estimation, which provides parameter estimates with the parametrized vector fields and their Lie derivatives needed at the approximately linearizing control law.15-18 In the case of r g 2, multilinear parameter products such as θˆ i2, θˆ iθˆ j, and θˆ iθˆ j appear at the higher more than one Lie derivatives. They are computed separately as if they are another parameters or computed by corresponding multilinear products of simple parameter estimates. And, this fact is also valid for the derivatives of time-varying parameters. Approximately linearizing feedback law obtained by substituting the estimated Lie derivatives into formulas (10) and (11) is applied to the design model, and then after some manipulation, the following quasi-linear error system results in

ej˙r+1 ) Acejr+1 + W(ejr+1,η,ua)Θ

(12)

where ejr+1 ) [σ,e1,..,er]T and Ac is a (r + 1) × (r + 1) Hurwitz matrix with polynomial coefficients, Rk, k ) 0,1,..,r, and ua is the approximately linearizing control law. Θ is the parameter error vector between the estimated parameters and the actual parameters including all multilinear parameters products, and the matrix, W(‚) consists of nonlinear functions before the parameter error. Since the second term in the righthand side is induced from uncertain parameters, as each parameter is approaching its true value, the parameter error vector is decaying to zero and then the quasi-linear system becomes asymptotically linearized. The stability of the perturbed linear system can be ensured if the perturbing term is bounded over the state domain during the transient of parameter estimation and the poles of the Hurwitz linear system are placed sufficiently deep into the left half of the s-plane.24 Remark 1. For cases 1 and 2, the disturbance model is formed but excluded from the design work. If the disturbance model is to be excluded, it should be stable under the closed-loop feedback control. Under inputoutput linearizing control, the disturbance model makes up the hidden dynamics, which are unobservable from the outputs and are known as zero dynamics.1 Therefore, for the stability of an overall closed-loop system, the hidden dynamics should be bounded input-to-state stable, which requires the system of (1) to be a minimumphase system1 and is one of inherent restrictions of the input-output linearizing control.

Remark 2. As in case of time-invariant systems, it is possible to find singular points where the finite relative order is ill-posed; i.e., (LgyLfk yhy)(y,t) ) 0 for all > 0. At these points, the controllability is lost and care must be taken to avoid these points at the controller design. In such cases, one has to resort to approximated linearization techniques such as extended linearization or pseudolinearization.25,26 These approaches can be used for the time interval over which the relative order of the time-varying system becomes infinite.10 3.2. Parameter Estimation. Uncertain parameters in the design model should be estimated on line and their estimates utilized in the linearizing control law to make robust the linearization of uncertain design model. Consider a slightly rearranged design model in which the parameter vector, θ(t) ∈ Rp, is factored out: p

∑ i)1

y˘ - {f y0(y) + gy0(y)u} ) {

p

f yi (y) +

gyi (y)u}θ(t) ) ∑ i)1 Ψ(y,u)θ(t) (13)

where Ψ(y,u) is a (q × p) parameter coefficient matrix and it is assumed that the rank of Ψ(y,u) is p over H(Γ) × Ω, which means that the number of estimable parameters should be less than or equal to the number of available outputs. This assumption yields the result that if the outputs and their derivatives are changing, the parameters are also accordingly changing. Thus, it is for the observability of parameters from the outputs and it is a sufficient condition for parameter estimation through the model inversion as shown below. As a design requirement, the parameter estimates are required to be estimated such that the estimated outputs reach their actual outputs along the specified path defined by a reference model. The estimated outputs, yˆ , are obtained from the design model with the parameters replaced by their estimates, θˆ (t) ∈ Rp p

yˆ˙ - {f y0(yˆ ) + gy0(yˆ )u} ) {

∑ i)1

p

f yi (yˆ ) +

gyi (yˆ )u}θˆ (t) ) ∑ i)1 Ψ(yˆ ,u)θˆ (t) (14)

At this work, the proportional integral (PI) trajectory of the estimated output error as a target for the first time derivative of each estimated output to follow is taken as a reference model as in the GMC:23

∫0t (y - yˆ ) dt′

yˆ˙ ) K1(y - yˆ ) + K2

(15)

where K1 and K2 are diagonal matrices whose elements as tuning parameters specify prior performance of the parameter estimation. The full column rankness of the parameter coefficient matrix, Ψ(yˆ ,u), and the linear parametrization allow analytical derivation of the formula for parameter estimation by combining eqs 14 with 15 and the resulting parameter estimator in state space is

∫0t (y - yˆ ) dt′

yˆ˙ ) K1(y - yˆ ) + K2

θˆ (t) ) {ΨTΨ}-1ΨT{K1(y - yˆ ) + K2

(16a)

∫0t (y - yˆ ) dt′ -

(f y0(yˆ ) + gy0(yˆ )u)} (16b)


As seen in the parameter estimator, the reference model plays the role of a linear filter for the available outputs; in the case of K1 * 0 and K2 ) 0, it is a first-order filter and in the case of K1,K2 * 0, it is a second-order filter with a lead term. Thus, parameter estimates of (16b) are calculated based on the filtered outputs. The substitution of estimated parameters of (16b) into (14) makes the estimated outputs and the estimated outputs error a second-order linear response, which corresponds to the reference model of (15):

e ) (Is2 + K1s + K2)-1Is2y

(17)

yˆ ) (Is2 + K1s + K2)-1(K1s + K2)y

(18)

where e ) y - yˆ is the estimated output error. Tuning parameters K1 and K2 can be selected as suggested by the GMC23 in which diagonal elements, k1i, k2i of each matrix, K1, K2 are set as

k1i ) 2ξi/τi

(19a)

k2i ) 1/τi2

(19b)

and ξi is chosen to give the desired shape of the reference model and τi is selected to obtain the appropriate speed of the reference model. To avoid undesirable overshoot and oscillation, ξi is advised to be set to be more than 3. Remark 3. The features of GMC-like parameter estimation are similar to those of the nonlinear model reference control (MRC).23,27 For control affine multivariable systems with the relative order of 1, the MRC produces the control law that forces the process to follow a desired reference model defining the path to the set point, and such a control law is directly derived via model inversion. The reference model is the same as that of the parameter estimation and a sufficient condition for model inversion is the full rankness of the characteristic matrix before the control input, which is parallel to that of the parameter coefficient matrix. The parameter estimation by Tatiraju and Soroush28 was based on the reference model of the first order, and its similarities to the control synthesis, not to MRC, was also mentioned; their parameter estimator is a left inverse of the process while a model-based controller is a right inverse of the process. Remark 4. A sufficient condition for parameter estimation is the observability of the parameters from the outputs. The loss of sufficient condition means that some parameters are not observable, and so, parameters cannot be calculated by the above method. In that case, one possible solution is to redefine the estimable parameters of the design model so that the full rankness of parameter coefficient matrix may be maintained over the interested domain. Another one is to search for parameter estimates through a minimization problem such that the estimated outputs follow the reference model as closely as possible as does in the MRC when one of the required conditions is lost.23,27,28 Remark 5. The inclusion of an integral term in the reference model and the full rankness of the parameter coefficient matrix force the parameter estimates to get to their true values if unknown disturbances and modeling errors affecting the outputs do not exist. Otherwise, since the estimated outputs approach the available outputs without any offset, the parameter

estimates will incorporate the effects caused by disturbances and modeling errors, and as a result, they show permanent deviation from the true values, which helps to improve the robustness of the controller because uncertainties are reflected into the linearizing control law in the form of estimated parameters. And, the existence of an integral term does not ask for large tuning values to eliminate any offset in the estimated output error, while in case of the reference model of the first order, large tuning values are required for offset free, which may tend to induce overshooting responses.29 Remark 6. If tuning parameters for parameter estimation are selected as suggested by the GMC23 and the rank deficiency of the parameter coefficient matrix can be suitably avoided on-line, the parameter errors will be bounded. And, since the parameters are introduced mainly as functions of the state of the disturbance model defined by smooth vector fields, their derivatives are also smooth and bounded. Thus, when the parameter estimates are used with the linearizing control law, the stability of the closed-loop quasi-linear system of (12) can be kept if the poles of the Hurwitz matrix, Ac are located sufficiently deep into the left half plane and the perturbation term induced by parameter error is bounded.24 Remark 7. The necessity of the state in the control synthesis1,9,14,19-22 and the parameter estimation,15-18 which is a common requisite for nonlinear system design, is eliminated. This is realized by the fact that unavailable states are lumped into estimable parameters and the reduced design model is described in the output variables as the state. Since the process model as an open-loop state observer is often open to diverse uncertainties and the design of a nonlinear closed-loop state observer is another challenging problem in the control practice, this feature is practically important. Remark 8. Equation 16a of the parameter estimator in state space is a second-order linear system driven by the available outputs and as well their first derivatives. To obtain approximate derivatives of the outputs available in time series, a differentiator filter (leadlag filter) can be used. Approximate output derivatives may show erratic behavior even for noiseless outputs, which in turn yield erratic parameter estimates. To remove this adverse effect, a first-order filter for filtering each estimated parameter is introduced and the filtered parameter estimates are used in the control law. The value of the filter constant within 0 eRf < 1 is usually set to be a larger one for noisy outputs. A larger filter constant gives more smooth but less accurate estimates while a smaller constant gives fluctuation tending but more accurate estimates. 4. Applications and Discussions To illustrate the applicability and the effectiveness of the proposed method, three typical control problems are considered: (a) the regulation of a continuous fermenter; (b) the tracking of an optimal temperature trajectory in a batch polymerization reactor; (c) the control of a chemical reaction system in a CSTR. Case a is the simplest since the relative order of the system is 1, which makes the design model be a scalar equation with a single parameter, but this case is effective to show clear-cut features of the method. Case b shows that applicable systems may be broad. Although a batch reactor is considered, slight modifications of the design model include the temperature control of a continuous

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and so inconsistent performances result. The ALIC does not require the state estimation, but instead, a parameter estimator is needed. The parameter estimator estimates unknown parameters from on-line measurements and provides the estimates with the control law. So, the ALIC gives consistent control performances if any changes in the actual process do not happen. The differentiator filter in the schematics of Figure 1 is for derivatives of the reference output at the output tracking problem. Its function at the tracking problem by the GLC was interpreted as a nonlinear bias of an external PI controller and the bias, vb is given as12 r

vb ) ysp +

Rky(k) ∑ sp k)1

(20)

This term is easily extracted from the linearizing control law of the ALIC in the error coordinate, and it enables the controller to take a priori action to track the reference trajectory. Two issues related to capabilities of the ALIC are considered: (a) measurement noises and (b) the existence of external disturbances unrelated to parameters to be estimated. 4.1. Regulation of a Continuous Fermenter. Consider a continuous fermenter with a sterile feed (Xf ) 0) and an effluent stream. The fermenter is well mixed and it has a constant volume. The state space model suitably describing this system is given as follows: 14,31

[]

[

][

]

µ(x2,x3,ϑ)x1 x˘ 1 -x1 1 x1µ(x2,x3,ϑ) + x2 - x2f u (21) x˘ 2 ) YX/S x˘ 3 -x3 (Rµ(x2,x3,ϑ) + β)x1

Figure 1. Schematics of three control schemes: (a) adaptive linearizing integral control (ALIC); (b) globally linearizing control (GLC); (c) nonlinear internal model control (NIMC).

chemical reactor. The relative order of the system is 2, and necessary outputs are normally available. Compared with cases a and b, case c is more or less obscure in the sense that the availability of (r - 1) secondary outputs is not clear. But necessary secondary output is estimated from the physical relationship of the system. The control scheme in Figure 1a, ALIC, is a combination of an input-output linearizing control and a parameter estimation. The control performance of the ALIC is compared with other control schemes; the conventional PI controller, the GLC in Figure 1b, and the NIMC in Figure 1c. Due to an integral term of the ALIC, the ALIC is equivalent to the GLC in the sense that the linearized systems can be made equal to each other. The GLC and the NIMC use the process model as a state observer, which has been a common practice in the chemical process control.30 Since the process model is inevitably exposed to diverse uncertainties and disturbances, the performances of the GLC and the NIMC depend on specific conditions of the process model

yc ) x1

where x1 ) X, x2 ) S, and x3 ) P are the cell mass concentration, the substrate concentration, and the product concentration, respectively. x2f is the substrate concentration in the feed and R and β are constant product yield parameters. µ and YX/S are the specific growth rate and the cell mass yield coefficient, and they are known to be sensitively varying to operating conditions. ϑ is the parameters set of the µ model exhibiting the substrate and the product inhibition:

µ)

µm(1 - P/Pm)S Km + S + S2/Ki

(22)

where µm is the maximum specific growth rate, Pm the product saturation constant, Km the substrate saturation constant, and Ki the substrate inhibition constant. The control objective is to maintain the cell mass concentration, yc ) x1, at the regulated point by manipulating the dilution rate, u ) D. Since the regulated point determined through a suitable optimization is near the washout region and the disturbances in the µm and the YX/S have a significant effect on the washout condition, a well-designed controller is required.14,31 Operating conditions and nominal parameters for simulation can be referred to the literature.14,31 The controlled output, x1, is assumed to be measurable. Secondary outputs for control synthesis are not needed because the system has the relative order of 1.


Since unavailable states, x2, x3, of the disturbance model can be used in the state transformation along with the controlled output, the transformed system is identical to the original system. Therefore, the design model consists of the cell mass balance equation. Because the specific growth rate, µ, has a functional form of unavailable states, it is taken as an unknown parameter. The design model in the output variable and also in a parametric form is

y˘ c ) -µ(x2,x3,ϑ)yc + (-yc)u ) -θ(t)yc + (-yc)u

(23)

The effects of the disturbance model on the design model are displayed through µ(‚). All assumptions necessary for the proposed method are satisfied unless the system is in a washout condition where yc ) 0. The control synthesis and the design of parameter estimator will proceed based on the design model of (23). This fermenter system was also taken as an example for an illustration of the NIMC14 and of the adaptive IMC.16 Some of their case studies are adopted here and tuning parameters of the linearizing control laws are also taken from their work, in which for the NIMC, the linearized system is made to be 1/(s + 1) and for the GLC, the linearized system is 1/(s + 1)2, with ) 1.0 for both. The ALIC is set to be the same as the GLC. The PI controller’s parameters are set as Kc ) -0.07 and τI ) 4.5, whose values are tuned such that the set point tracking performances are similar to those of the linearizing control result.14,16 The tuning parameters for parameter estimation are set as ξ ) 5 and τ ) 0.5 in eqs 19a,b. The constant of the first-order filter for filtering the estimated parameters is set to 0.95, and the sampling time for control and parameter estimation is 1 min. As is often the case, the tuning parameters are referenced to the open-loop dynamics from step test of the dilution rate. Figure 2a shows the disturbance rejection capabilities of four controllers when a step disturbance of -20% in the µm is introduced. Since the disturbance has the same relative order of 1 as the control input and it is not measurable, it cannot be rejected perfectly even though the full state is available. But, since the ALIC uses the estimate of this disturbance by way of the parameter, it gives the best rejection capability, which comes from the good results of the parameter estimation as shown in Figure 3. The estimated parameter reaches the actual one smoothly in a short time. The control input moves by each controller are comparable as shown in Figure 2b. The GLC gives better control performance than the NIMC, and the PI controller yields the poorest performance. This result is as usual, but has an exception. Figure 4 shows the control performances of four controllers when a step disturbance of -20% in the YX/S occurs. Since the disturbance relative order (F ) 2) is larger than the input relative order (r ) 1), if exact linearization can be achieved, i.e., the full state and the right model for µ are available, this disturbance can be perfectly rejected without its measurement. Because closely approximated linearization is achieved through the estimated parameter accounting for the disturbance effect, the ALIC almost perfectly rejects the disturbance. But since the GLC and the NIMC use the state from the process model, they cannot effectively reject the disturbance. The PI controller shows the poor performance. Figure 5a shows the control performances of four controllers when a sudden nonsterile feed (Xf ) 0.6) is

Figure 2. Disturbance rejection capabilities of four controllers when a step disturbance of -20% in µm is introduced: (a) cell mass concentration, X; (b) control input moves of the dilution rate, D.

Figure 3. Estimated parameter and actual parameter, specific growth rate, µ(‚), when a step disturbance of -20% in µm is introduced.

fed. This case was not studied in the literature.14,16 Since this disturbance has the relative order of 1 and it is unmeasured, it cannot be perfectly rejected. Although it is an external disturbance not directly related to µ, its effects on the output are reflected into the parameter estimate, µˆ , which makes a difference between the estimated parameter and the actual parameter as shown in Figure 5b. But, since the ALIC can compensate for the disturbance through the deviated parameter estimate, it shows superior disturbance rejection capability. In contrast, the GLC and the NIMC give poor performances, and exceptionally, the NIMC yields a poorer performance than the PI controller. To illustrate the measurement noise effect, the white noises of the standard deviation of 0.01 are imposed on the output measurements and subsequent filtering of the measured output is not done. As shown in Figure

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Ind. Eng. Chem. Res., Vol. 41, No. 21, 2002 Table 1. Reaction Conditions for a Polymerization Batch Reactor variablea

Figure 4. Disturbance rejection capabilities of four controllers when a step disturbance of -20% in YX/S is introduced.

VoS VoM VoR VJ Tcw Ao CoI CoM CoS Uo tf xd PDId MWnd

description

value

solvent volume monomer volume reaction volume jacket volume coolant temperature area for heat transfer initiator concentration monomer concentration solvent concentration overall heat-transfer coeff batch cycle time desired monomer conversion desired polydispersity index desired number-average MW

0.3 (L) 1.0 (L) 1.3 (L) 0.5VoR 10 (°C) 500 (cm2) 0.0258 (g-mol/L 6.9675 (g-mol/L) 2.1019 (g-mol/L) 0.75 (cal/min cm2 K) 380 (min) 1.0 2.5 360 000 (g/mol)

a Superscript o means the value at initial condition or at no conversion.

methacrylate (MMA) polymerization batch reactor is considered. The temperature trajectory is determined off-line through the minimization of the weightedsquared sum of each deviation of the product qualities expressed in terms of the conversion, the numberaveraged molecular weight, and the polydispersity index (PDI) from the specified value at the final batch time.32,33 Reaction kinetics of a free-radical MMA polymerization is relatively well known,34 and all describing equations are appropriately nondimensionalized to prevent numerical conditioning problems. The reaction conditions are tabulated in Table 1, and related physical properties and reaction rates constants can be referred to the literature.34 The design model for temperature trajectory tracking is an energy balance around the reactor:

T h˙ R ) θ h 1(x,λh0,V h ,T h R,ϑ) - θ h 2(x)(T hR - T h J) T h˙ J ) γθ h 2(x)(T hR - T h J) + F h cw(T h cw - T h J) + q j In (24)

Figure 5. Disturbance rejection capabilities of four controllers when a nonsterile feed (Xf ) 0.6) as an external disturbance is fed: (a) cell mass concentration, X; (b) estimated parameter and actual parameter, specific growth rate, µ(‚).

6, the adverse effects of the noises on the parameter estimation and the control performance of the ALIC are not noticeable when a step disturbance of -20% in the µm is introduced. Case studies14,16 such as a structural model error in the growth rate, µ, and unmodeled dynamics for the disturbance model were performed. Because the NIMC and the GLC need an appropriate model as a state observer, the studies are meaningful in showing the their robustness. But, the ALIC is independent of the process model, and it gives consistent performances unless some changes happen to the actual process, which is one of the advantages of the ALIC. 4.2. Tracking of an Optimal Temperature Trajectory in a Batch Polymerization Reactor. Tracking of an optimal temperature trajectory in a methyl

where the parameters, θ h 1(‚) and θ h 2(‚) represent the heat of reaction and an overall heat-transfer coefficient, h , ϑ, qj in, respectively. γ is relatively constant, and x, λh0, V h cw are the monomer conversion, the total F h cw, and T concentration of polymer radicals, the reaction volume, the process parameters set, the heat input from the heater, the coolant flow rate, and coolant temperature, respectively. Each parameter depends on the states(x, h ) of the disturbance model, which consists of the λh0, V material balances based on the reaction kinetics and of which the states are usually difficult to measure. If the involved states are not estimated by some relevant h J are methods, the parameters are unknown. T h R and T the temperatures in the reactor and in the jacket side, and both are measured outputs. The polymerization reaction is highly exothermic and also highly nonlinear mainly because of the gel effect and glass effect,35 which give rise to an aggressive heat production during the course of the reaction and hamper the completion of the reaction. As the reaction proceeds and more polymers are formed, the reaction medium gets more viscose and severe heat-transfer restrictions occur. To incorporate these facts into the simulated process, the gel and glass effects model of Chiu et al.36 and the empirical correlation for the heat transfer coefficient by Soroush and Kravaris12 given as a function of the conversion. Each parameter has a clear physical


Figure 6. Measurement noise effects on the performances of the ALIC: (a) cell mass concentration, X; (b) estimated parameter and actual parameter, specific growth rate, µ(‚).

meaning and practical importance, but it is infeasible to know them a priori. The design model in the output variables can be written as

[] [

] []

y˘ 1 θ h 1(t) - θ h 2(t)(y1 - y2) 0 + u j y˘ 2 ) γθ h 2(t)(y1 - y2) 1 c

(25)

h cw(T h cw - y2) + q j In u jc ) F

(26)

yc ) [1 0 ]y where y ) [y1,y2]T ) [T h R,T h J]T and u j c means the net heat input to the batch reactor. Physical implementation of u j c is done by a coordination rule.12 The control objective is to force the reaction temperature to track a predetermined trajectory by manipulating the net heat input, u j c. When the parameters are taken as unknown, the design model has a relative order of 2. The linearly

parametric design model can be expressed as

[] [

][ ] [ ]

y˘ h1 1 -(y1 - y2) θ 0 jc y˘ ) y˘ 1 ) h2 + 1 u 0 γ(y1 - y2) θ 2

(27)

in which the parameter coefficient matrix, Ψ(y,u) is nonsingular unless the reaction temperature is equalized to the jacket temperature. The tuning parameters of the ALIC are determined via the design of the GLC. As suggested in the design procedure of the GLC,9,12 the system after linearizing is adjusted to the open-loop dynamics from step test, i.e., 1/(s + 1)2 with ) 0.01, and by a standard tuning procedure of a linear system,37 an external PI controller with Kc ) - 1. × 105 and τI ) 2 is designed around the linearized system. Then, the polynomial coefficients of the ALIC are set such that both feedback systems are equivalent to each other. The reference dynamics for tuning parameters are obtained from step test of the coolant flow rate under the condition that the reaction does not occur and a constant external heat input is

5256


Figure 8. Estimated parameters and actual parameters: (a) heat of polymerization reaction, θ h 1(‚); (b) overall heat-transfer coefficient, θ h 2(‚).

Figure 7. (a) Tracking performances of a reference reaction temperature trajectory by three controllers; (b) jacket-side temperatures; (c) coolant flow rates.

applied. Since the system has the relative order of 2, the approximately linearizing control law contains the first derivatives of the estimated parameters, but the derivatives are ignored in the applied control law. The parameters of PI controller used for comparison are set as Kc ) -1.1 × 103 and τI ) 2, whose values are tuned to give a tracking performance similar to those of the linearizing control at the upward and downward part of the bell-shaped trajectory. Parameters for parameter estimation are set as ξ ) 5 and τ ) 0.01 in eqs 19a,b. The filter constant of the first-order filter for filtering each parameter estimate is set to 0.95, and the sampling time for control and parameter estimation is 2.2 × 10-4 (5 s) in dimensionless time. Figure 7a shows the tracking performances by three control methods, the ALIC, the exactly linearized GLC, and the PI control, and panels b and c of Figure 7 show the jacket temperature and the control input moves by each controller. If exact parameters are available, exact linearization in the input-output response is possible,

and it can be played as a basis for comparisons. The tuning parameters of the exactly linearized GLC seem to be reasonable because there are good tracking results and the least control moves are consumed. The control performance of the ALIC will depend on the results of the parameter estimation. As shown in Figure 8, the estimates for both parameters agree with their true values despite a peaking in the reaction heat during the gel effect. Therefore, the control performance and the control input moves are comparable to those of the exactly linearized GLC. Despite big and steep input moves, the parameter estimates show rather smooth behaviors. As expected, the PI controller gives poor performances and it requires larger control input. At the end of the batch time, big fluctuations appear to be continuing. In the usual GLC,9,12 the disturbance model as a state observer is subject to diverse realistic situations, which leads to provision of inappropriate knowledge with the GLC controller and so limits the right control actions on time. This fact can be confirmed from the predicted parameters in relation to their actual values. In the case of a +20% error in the initiator loading, the predicted parameters precede the actual parameters in advance and so early deviations occur at the head of the trajectory because a prior control action of increasing the coolant flow rate is applied (Figure 9). In the case of a -20% initiator loading error, since the predicted parameters do not reflect the magnitude of actual parameters, insufficient and out-of-time control inputs are applied, and as a result, low reaction temperature is induced like the PI controller at the end of the batch time (Figure 9). For kinetic parameter errors in the frequency factors of the propagation reaction and the


For this reaction system, it is desired to control the concentration of the product C, x3, at the regulated point by the molar feed rate of the reactant B, u. Thus, the controlled output is yc ) x3 and it is assumed to be measurable. But, the output controllability cannot be achieved through the design model formed from only this output. So, additional outputs satisfying the required assumptions should be found out. Because the system has the relative order of 2, at least one more suitable output is required. The following mole balance based on the reaction stoichiometry is valid at steady state Figure 9. Tracking performances of a reference reaction temperature trajectory by the ALIC and the GLC for initiator loading errors.

initiator decomposition reaction, the behaviors of the predicted parameters and the tracking performances are similar to those of the case of initiator loading error. The noise effects of the output measurements and the existence of external disturbances, are also considered. Random noises of a standard deviation of 0.01 for the reaction temperature and of a standard deviation of 0.015 for the jacket temperature are imposed on the respective measurements, any filtering for noisy measurements is not done, and tuning parameters of the parameter estimator are unchanged. As shown in Figure 10, the estimated heat-transfer coefficient shows some fluctuations coming from the fluctuating control input before the gel effect because the control input is involved in the parameter estimation and the control law is tightly tuned for tracking the downward part of the trajectory. On the contrary, noise effects on the estimates of the heat of polymerization reaction are slight. Despite parameter fluctuations, the control performances of the ALIC do not show noticeable deterioration. The heat loss to the surroundings from the reactor jacket is taken into account in the jacket-side energy balance as an external disturbance. There is a noticeable deviation, which corresponds to the heat loss, in the estimated overall heat-transfer coefficient, but since the estimates are directly used in the control law, the performance of the ALIC is unchanged against the external disturbance. 4.3. Control of the Chemical Reaction System in a CSTR. Consider a continuous perfectly mixed stirred tank reactor (CSTR) in which a successive reaction (A T B f C) takes place under isothermal conditions (Figure 11)19,20 with the rates of reaction given by

r1 ) k1CA - k2CB2 r2 ) k3CB2

(28)

and described by the dimensionless equations

x˘ 1 ) 1 - x1 - Da1x1 + Da2x22 x˘ 2 ) Da1x1 - x2 - Da2x22 - Da3x22 + u

(29)

x˘ 3 ) Da3x22 - x3 The variables and parameters in the equations can be referred to the literature,19,20 and the nominal parameter values are Da1 ) 3.0, Da2 ) 0.5, and Da3 ) 1.0.

CAfF + NBf ) (CA + CB + CC)F

(30)

and in the dimensionless variables it is

1+u j ) xj1 + xj2 + xj3.

(31)

The bar over the variables denotes those at steady state. The relationship is valid under transients if the reaction rates are instantaneous. Otherwise, a dynamic relationship should be needed, which is obtained by the direct summation of the states equation:

w˘ ) -w + u + 1

(32)

where w ) x1 + x2 + x3. For convenience, this equation will be written in the deviation variables

w ˜˙ ) -w ˜ + u˜

(33)

where w ˜ )w-w j with w j ) xj1 + xj2 + xj3 and u˜ ) u - u j. The steady-state values as a basis are determined with u j ) 1 and nominal parameters: xj1 ) 0.3467, xj2 ) 0.8796, and xj3 ) 0.7737. The w ˜ is driven by the control input and an initial condition. That is, the total moles of all components in the reactor is determined by the molar rate of injected reactant B and initial total moles in the reactor. If the secondary output is taken as yˆ 2 ) w ˜ +w j - yc, it becomes equal to ys ) x1+x2 algebraically. When the initial of w ˜ , i.e., the direct sum of two states, x1 and x2, at the initial point is known, yˆ 2 gives exactly ys because w ˜ is driven by a known input. If the initial of w ˜ is unknown, yˆ 2 is not equal to ys during the initial transient, but the output error between yˆ 2 and ys exponentially decays to zero. Once the error gets to zero, the estimated output, yˆ 2 is exactly ys. Therefore, the estimate, yˆ 2, with eq 33 is useful for obtaining the secondary output, ys. In the design afterward, the output, ys ) x1 + x2, is assumed to be available because its value for control and parameter estimation can be taken from yˆ 2. The argument above is not valid if there are unknown disturbances and modeling errors, but the case that is still valid under an unknown side reaction will be discussed at the end of this subsection. With the help of the secondary output, ys ) x1 + x2, the augmented output vector becomes y ) [yc,ys]T ) [y1,y2]T and the state transformation can be taken as

[] [ ] x3 y1 y2 ) x1 + x2 x1 η

(34)

5258


Figure 10. Measurement noise effects on the estimated parameters: (a) heat of polymerization reaction, θ h 1(‚); (b) overall heat-transfer coefficient, θ h 2(‚).

modified by substituting the relation x2 ) y2 - x1 into the upper equation of the model. Then, the modified model is

y˘ 1 ) Da3y22 - Da3θ1(t) - y1 y˘ 2 ) 1. - y2 - Da3θ2(t) + u Figure 11. Schematic of chemical reaction system in a CSTR.

The design model in the output variables through the state transformation is

y˘ 1 ) Da3x22 - y1 y˘ 2 ) 1 - y2 - Da3x22 + u

(35)

This model satisfies assumption 2(a), but because the relative order is not defined, the design model is

(36)

where θ1(t) ) 2x1y2 - x12 and θ2(t) ) x22 are introduced as unknown parameters due to their dependences on unavailable states. The relative order of the controlled output is defined for this design model and it is 2. Therefore, the design model in the output variables for control synthesis is

[] [

] []

y˘ 1 Da3y22 - Da3θ1(t) - y1 0 ) + u y˘ 2 1 1 - y2 - Da3θ2(t) yc ) [1 0 ]y

(37)


Figure 13. Estimated parameters and actual parameters introduced at eq 36 when eq 33 has an exact initial value.

Figure 12. Control performances of three controllers when the operating point is moved to new one: (a) concentration of the product C, x3; (b) molar feed rate of the reactant B, u.

and its parametric form for parameter estimation is

[] [

] [

][ ]

y˘ 1 Da3 0 θ1(t) Da3y22 - y1 ) Da3 θ2(t) y˘ 2 1 - y2 + u 0

(38)

Note that the parameter coefficient matrix is nonsingular if the parameter, Da3, is not zero, which is physically the case. So, the proposed method is applicable. The linearizing control law is designed such that the linear system after linearizing is 1/(s + 1)2 for both the GLC and the ALIC with ) 0.5. The external PI controller in the GLC and the integral term (R0 ) 0.) in the ALIC are not included and so both the GLC and the ALIC are reduced to the NIMC. Since the system has the relative order of 2, the first derivatives of estimated parameters needed in the control law are ignored as in the case of the second example. The PI controller parameters are set as Kc ) -2.0 and τI ) 0.5, whose values give a control speed similar to those of the linearizing controls at the start. Tuning parameters for parameter estimation are set as ξ ) 5 and τ ) 0.1. The filter constant for filtering parameter estimates is set as 0.8, and sampling time for control and parameter estimation is 0.1 unit residence time. All tuning parameters are determined by referring to the open-loop dynamics from the step test in the control input. Figure 12 shows the control performances and the control moves by three controllers when the operating point is moved from the initial steady state (u j ) 0.5196, xj1 ) 0.3125, xj2 ) 0.7071, xj3 ) 0.5) to the base steady state. The exact initial value for w ˜ is assumed to be available for the ALIC and the GLC for comparison is subject to (20% error in the initial value of the process

Figure 14. Estimated parameters and actual parameters introduced at eq 36 when eq 33 has an initial error and a side reaction occurs.

model used as a state observer. The ALIC gives the best control performance and the most reasonable control moves, which results from good agreement of the parameter estimates with actual parameters as shown in Figure 13, where both parameter estimates approach their actual parameters smoothly and fast. The GLC has some aggressive control moves at the beginning, which may be caused by incorrect initials, and furthermore, it gives inconsistent responses: fast and a little overshoot response for +20% initial error while initially fast but sluggish approach to the set point for -20% initial error. The PI controller shows poorer responses with some overshoot and oscillations. When an initial value for w ˜ is uncertain, an incorrect initial value produces an overshoot mainly in the parameter estimate, θˆ 1, which is caused by the difference of the estimated output, yˆ 2, from true secondary output, ys, during the initial transients. But, since the estimated output error exponentially decays to zero, the estimate reaches the true parameter fast as in Figure 14, and so its adverse effect on the performance of the ALIC is slight. Figure 14 includes the effect of a side reaction as an external disturbance, but the trends of parameter estimates at the initial transient are not different. Noise effects of the measurements, y1, on the parameter estimator and on the control performance of the ALIC are minor, as in the previous examples. In the work of Kravaris and Palanki,19 it is assumed that there is unmodeled first-order side reaction from B to demonstrate the robustness of their nonlinear state feedback. When a side reaction of B f D occurs, the ˜ +w j - yc is algebraically equal estimated output, yˆ 2 ) w to ys ) x1 + x2 + x4, not to ys ) x1 + x2, where x4 is the

5260


concentration of side product, D. So, instead of ys ) x1 + x2, the secondary output of ys ) x1 + x2 + x4 should be assumed to be available and it should be included into the state transformation. But, the design model resulted from the modified state transformation is the same as the eq 36 except for the details of introduced parameters. That is, the detailed θ1(t) is obtained by the substitution of x2 ) y2 - x1 - x4 into the upper equation of (35), but the θ2(t) remains unchanged. So, the effect of the side reaction as an unknown disturbance is exhibited through the parameter, θ1(t). Fortunately, the output, yˆ 2, estimated from (33) is equal to ys ) x1 + x2 + x4 despite a side reaction, and so parameter estimate, θˆ 1(t), becomes equal to its true value without any consistent deviation. In Figure 14, the value of θ1(t) is larger than that of θ2(t) at steady state, which is the reversal of the relative size between two parameters in the absence of this disturbance as seen in Figure 13. The degradations in the control performance of the ALIC caused by the side reaction are negligible because the estimated parameters are utilized in the control law. 5. Conclusions It was illustrated that the proposed method could fit diverse control problems in the chemical processes, and by simulation, it was demonstrated that the method could be applied to them with success, which results from good agreement of the parameter estimates by the GMC-like parameter estimation with actual values. For each example, the resulting design model is in a simple form for the complexities of the process considered. The design procedure for both control synthesis and parameter estimation is simple and clear due to the linearity of the design equation on which the selection of tuning parameters is based and is imposed. The robustness of the ALIC is established by the fact that diverse uncertainties and disturbances affecting the available outputs are reflected into the control law through the parameter estimates. The GMC-like parameter estimator turns out to be suitable for making robust the asymptotic inputoutput linearization. The proposed method requires only the output measurements and a parametrized reduced design model as the least knowledge on the given process, and the necessity of the full-state and an exact nonlinear model, which are the critical weak points of the state feedback linearization, are eliminated. Compared with the ALIC, the GLC and the NIMC that have been widely accepted in the nonlinear control of chemical processes may give poorer and inconsistent performances as shown in the simulation results. Acknowledgment We thank the Brain Korea 21 Program supported by the Ministry of Education and the National Research Lab Grant of the Ministry of Science & Technology for financial aid to this research. Literature Cited (1) Isidori, A. Nonlinear Control Systems, 2nd; Springer-Verlag: Heidelberg, 1989. (2) Jakubcyzk, B.; Respondek, W. On Linearization of Control Systems. Bull. Acad. Pol. Sci. (Math.) 1980, 28, 517. (3) Su, R. On the Linear Equivalents of Nonlinear Systems. Syst. Control Lett. 1982, 2, 48. (4) Freund, E. The Structure of Decoupled Nonlinear Systems. Int. J. Control 1975, 21, 443.

(5) Isidori, A.; Ruberti, A. On the Synthesis of Linear InputOutput Response for Nonlinear Systems. Syst. Control Lett. 1984, 4, 17. (6) Calvet, J.; Arkun, Y. Feedforward and Feedback Linearization of Nonlinear Systems and its Implementation Using Internal Model Control (IMC). Ind. Eng. Chem. Res. 1988, 27, 1822. (7) Henson, M. A.; Seborg, D. E. Adaptive Control of pH Neutralization Process. IEEE Trans. Control Syst. Technol. 1994, 2, 169. (8) Kam, K. W.; Tade, M. O. Nonlinear Control of a Simulated Industrial Evaporation System Using a Feedback Linearization Technique with a State Observer. Ind. Eng. Chem. Res. 1999, 38, 2995. (9) Kravaris, C.; Chung, C. B. Nonlinear State Feedback Synthesis by Global Input/Output Linearization. AIChE J. 1987, 33, 592. (10) Palanki, S.; Kravaris, C. Controller Synthesis for TimeVarying Systems by Input/Output Linearization. Comput. Chem. Eng. 1997, 21, 891. (11) Daoutidis, P.; Kravaris, C. Synthesis of Feedforward State Feedback Controllers for Nonlinear Processes. AIChE J. 1989, 35, 1602. (12) Soroush, M.; Kravaris, C. Nonlinear Control of a Batch Polymerization Reactor: an Experimental Study. AIChE J. 1992, 39, 1429. (13) Soroush, M.; Kravaris, C. Multivariable Nonlinear Control of a Continuous Polymerization Reactor: an Experimental Study. AIChE J. 1993, 39, 1920. (14) Henson, M. A.; Seborg, D. E. An Internal Model Control Strategy for Nonlinear Systems. AIChE J. 1991, 37, 1065. (15) Sastry, S.; Isidori, A. Adaptive Control of Linearizable Systems. IEEE Trans. Autom. Control 1989, 34, 1123. (16) Hu, Q.; Rangaiah, G. P. Adaptive Internal Model Control of Nonlinear Processes. Chem. Eng. Sci. 1999, 54, 1205. (17) Teel, A.; Kadiyala, R.; Kokotovic, P.; Sastry, S. Indirect Techniques for Adaptive Input-Output Linearization of Nonlinear Systems. Int. J. Control 1991, 53, 193. (18) Iyer, N. M.; Farell, A. E. Adaptive Input-Output Linearizing Control of a Continuous Stirred Tank Control. Comput. Chem. Eng. 1995, 19, 575. (19) Kravaris, C.; Palanki, S. Robust Nonlinear State Feedback Under Structured Uncertainty. AIChE J. 1988, 34, 1119. (20) Arkun, Y.; Calvet, J. Robust Stabilization of Input-Output Linearizable Systems under Uncertainty and Disturbances. AIChE J. 1992, 38, 1145. (21) Chou, Y.; Wu, W. Robust Controller Design for Uncertain Nonlinear Systems via Feedback Linearization. Chem. Eng. Sci. 1995, 50, 1429. (22) Kolavennu, S.; Palanki, S.; Cockburn, J. C. Robust Control of I/O linearizable Systems via Multi-model H2/H∞ Synthesis. Chem. Eng. Sci. 2000, 55, 1583. (23) Lee, P. L.; Sullivan, G. R. Generic Model Control(GMC). Comput. Chem. Eng. 1988, 12, 573. (24) Zak, S. H. On the Stabilization and Observation of Nonlinear/Uncertain Dynamic Systems. IEEE Trans. Autom. Control 1990, 35, 604. (25) Reboulet, C.; Champetier, C. A New Method for Linearizing Nonlinear Systems: the Pseudolinearization. Int. J. Control 1984, 40, 631. (26) Wang, J.; Rugh, W. J. On the Pseudolinearization Problem for Nonlinear Systems. Syst. Control Lett. 1989, 12, 161. (27) Boye, A. J.; Brogan, W. L. A Nonlinear System Controller. Int. J. Control 1986, 44, 1209. (28) Tatiraju, S.; Soroush, M. Parameter Estimator Design with Application to a Chemical Reactor. Ind. Eng. Chem. Res. 1998, 37, 455. (29) Jayadeva, B.; Rao, Y. S. N. M.; Chidambaram, M.; Madhavan, K. P. Nonlinear Controller for pH Process. Comput. Chem. Eng. 1990, 14, 917. (30) Daoutidis, P.; Kravaris, C. Dynamic Output Feedback Control of Minimum-Phase Nonlinear Processes. Chem. Eng. Sci. 1992, 47, 837. (31) Henson, M. A.; Seborg, D. E. Nonlinear Control Strategies for Continuous Fermenters. Chem. Eng. Sci. 1992, 47, 821.

Ind. Eng. Chem. Res., Vol. 41, No. 21, 2002 5261 (32) Thomas, I. M.; Kiparissides, C. Computation of the NearOptimal Temperature and Initiator Polices for a Batch Polymerization Reactors. Can. J. Chem. Eng. 1984, 62, 284. (33) Soroush, M.; Kravaris, C. Optimal Design and Operation of Batch Reactors. 2.A Case Study. Ind. Eng. Chem. Res. 1993, 32, 882. (34) Louie, B. M.; Carratt, G. M.; Soong, D. S. Modeling the Free Radical Solution and Bulk Polymerization of MethyMethaAcrylate. J. Appl. Polym. Sci. 1985, 30, 3985. (35) Ballagou, P. E.; Soong, D. S. Major Factors Contributing to the Nonlinear Kinetics of Free-Radical Polymerization. Chem. Eng. Sci. 1985, 40, 75.

(36) Chiu, W. Y.; Carratt, G. M.; Soong, D. S. A Computer Model for the Gel Effect in Free-Radical Polymerization. Macromolecules 1983, 16, 348. (37) Morari, M.; Zafiriou, E. Robust Process Control; Prentice Hall: Upper Saddle River, NJ, 1989.

Received for review December 30, 2001 Revised manuscript received June 28, 2002 Accepted August 2, 2002 IE011042E

Adaptive Linearizing Control of a Nonlinear Chemical Process Based

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